收敛加速与求和过程中的不规则输入数据：一般性考虑及若干特殊高斯超几何级数作为模型问题
Irregular Input Data in Convergence Acceleration and Summation Processes: General Considerations and Some Special Gaussian Hypergeometric Series as Model Problems
AI总结：
针对序列变换中前导项不规则导致结果不可靠的问题，以高斯超几何级数为例，提出排除前导项可改进求和，并推导新递推公式。
摘要：
序列变换通过检测并利用待变换序列元素的规律性来实现收敛加速或发散情况下的求和。对于足够大的指标，通常确实存在某些渐近规律性，但序列的前导元素可能表现得相当不规则。高斯超几何级数 2F1 (a, b; c; z) 非常适合阐明此类问题。对于大多数参数和自变量，序列变换表现良好。然而，如果非终止超几何级数 2F1 的第三个参数 $c$ 是负实数，则各项初始大小增长类似于轻度发散级数的项。将此类级数的前导项用作输入数据会导致不可靠甚至完全无意义的结果。相比之下，如果在变换过程中排除前导不规则项，序列变换则能产生良好结果。类似问题也出现在微扰展开中。例如，通过从变换过程中排除前导项，可以显著改进六次非谐振子在无穷耦合极限 k_3 下的求和结果。最后，推导出了大量关于 2F1 (a, b; c; z) 的新递推公式。
英文摘要：
Sequence transformations accomplish an acceleration of convergence or a summation in the case of divergence by detecting and utilizing regularities of the elements of the sequence to be transformed. For sufficiently large indices, certain asymptotic regularities normally do exist, but the leading elements of a sequence may behave quite irregularly. The Gaussian hypergeometric series 2F1 (a, b; c; z) is well suited to illuminate problems of that kind. Sequence transformations perform quite well for most parameters and arguments. If, however, the third parameter $c$ of a nonterminating hypergeometric series 2F1 is a negative real number, the terms initially grow in magnitude like the terms of a mildly divergent series. The use of the leading terms of such a series as input data leads to unreliable and even completely nonsensical results. In contrast, sequence transformations produce good results if the leading irregular terms are excluded from the transformation process. Similar problems occur also in perturbation expansions. For example, summation results for the infinite coupling limit k_3 of the sextic anharmonic oscillator can be improved considerably by excluding the leading terms from the transformation process. Finally, numerous new recurrence formulas for the 2F1 (a, b; c; z) are derived.
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